Seismic response of mountain tunnel induced by fault slip

With the rapid development of Chinese transportation networks, such as the Sichuan-Tibet railway, numerous tunnels are under construction or planned in mountainous regions. Some of these tunnels must traverse or be situated near active fault zones, which could suffer damage from fault slip. In this study, the seismic response of a mountain tunnel subjected to coseismic faulting was analyzed using a fault-structure system in a two-step process. Firstly, a nonuniform slip model was proposed to calculate the ground deformations and internal displacements induced by a specific active fault on a geological scale, considering nonuniform slips on the fault plane. The 1989 Loma Prieta and 2022 Menyuan earthquakes were chosen as case studies to validate the proposed slip model. Secondly, the calculated displacement of the Menyuan earthquake was used as the input load for the discrete–continuous coupling analysis of the Daliang tunnel on an engineering scale. The simulated deformation of the Daliang tunnel aligned with the on-site damage observations following the Menyuan earthquake. Lastly, the effects of different fault conditions on the tunnel seismic response were investigated. The results indicate that the distribution of the peak longitudinal strain of the lining is governed by fault mechanisms, and the degree of fault slip significantly influences the response of the tunnel. A tunnel passing through an active fault with a wider fault fracture zone and smaller dip angle experience less damage.

www.nature.com/scientificreports/ of underground structures involves three major steps: (1) defining the seismic environment and developing seismic parameters; (2) evaluating the ground response to shaking, including ground failure and deformations; and (3) assessing the structural behavior due to seismic shaking.The key challenge is to evaluate the seismic loads subjected to underground structures.In experimental model tests and numerical modeling, fault slip is usually assumed to be a constant, which is determined by empirical relationships 26 .However, fault plane slip is heterogeneous, causing spatially varied ground deformation [27][28][29] .Okada's theories 30,31 are widely used to obtain spatially varied ground deformation due to fault slip.although they assume constant slip on the fault, differing from real seismic events inversion.
In this study, the seismic response of a tunnel due to coseismic faulting is analyzed using a fault-structure system.First, a nonuniform fault slip model is proposed to estimate the surface and internal deformation in Section "Surface and internal deformation induced by nonuniform fault slip".The Loma Prieta and 2022 Menyuan earthquakes are selected to validate the nonuniform fault slip model.Then, the seismic response of the Daliang tunnel due to coseismic faulting is analyzed using discrete-continuous coupled numerical simulation in Section "Seismic response of the Daliang tunnel caused by spatial varied internal displacement".Finally, the effects of different fault conditions and parameters on the tunnel coseismic response of due to fault slip are discussed.

Surface and internal deformation induced by nonuniform fault slip
To consider the spatially varied slip on the fault, a source model with a nonuniform slip distribution is proposed.The Loma Prieta and 2022 Menyuan earthquakes are selected as case studies to validate the proposed slip model.

Method
Okada 30,31 proposed a comprehensive set of analytical expressions for surface and internal displacements, strains, and tilts owing to inclined shear and tensile faults in a half-space for both point and finite rectangular sources.As shown in Fig. 1, the x-direction is parallel to the strike of the fault, the y-direction is perpendicular to the strike, and the z-direction is vertical.The origin of coordinate O is located on the surface, which is the vertical projection of the lower left point on the footwall plane.δ, W, and L are the dip angle, width, and length of the fault plane, respectively.U 1 , U 2 , and U 3 are the average dislocation components in the strike, dip, and tension directions, respectively.λ and μ are Lame's constants of the underground medium.Surface displacement induced by fault slip can be determined by Okada's solutions 30,31 .

Uniform slip model of asperity
Seismologists define an asperity as a fault rupture region with higher slip relative to the average slip on the fault.Somerville et al. 27 used a rectangular definition for asperities to facilitate slip model generation for future earthquakes.The average slip on the asperity is 2.01 times the slip averaged over the entire fault rupture surface.
In the asperity model, the fault plane is divided into several subfaults of equal area, as shown in Fig. 2. L and W represent the length and width of the entire fault plane, respectively, while dl and dw are the length and width of the subfaults, respectively.For earthquake scenarios, the source parameters can be obtained as follows: www.nature.com/scientificreports/According to Aki and Richards 37 , the average slip u on the fault plane is determined by where M 0 denotes the seismic moment, µ denotes the shear modulus, and A denotes the rupture area of the fault plane.The relationship between the seismic moment M 0 and the moment magnitude M w is 32 : and substituting Eq. (1) into Eq.( 2) yields: As shown in Eq. ( 3), the average slip of the fault is related to its moment magnitude, shear modulus, and rupture area.In Wells and Coppersmith' empirical relations 26 , the fault rupture area and fault length are related to the seismic magnitude as follows: Subsequently, the traditional asperity model with uniform slip can be determined using the following procedures 27 : (1) Two asperities are set on the entire fault plane: one large and one small.
(2) The areas of the large and small asperities are A as1 = 0.16LW and A as2 = 0.06LW , respectively.
(3) The average slip of the asperities u as is approximately 2.01 times the value of the entire fault plane slip u as =2.01u.(4) The average slip of the background area u b is approximately 0.71 times the value of the entire fault plane slip u b = 0.71u.
The surface and internal deformation induced by the slip of subfaults in the asperity and background regions can be calculated using Okada's theories 30,31 .The total seismic moment of the asperities and background area equals the seismic moment of the target fault: where M 0 ,M 0as1 ,M 0as2 , and M 0b denote the seismic moments of the entire fault, large asperity, small asperity, and background, respectively.

Nonuniform source slip model
The definition of asperities can consider slip variation over the entire fault plane.However, slip variation within the asperity region cannot be account for.Moreover, the slips on the edge of the asperity and the adjacent background area exhibit a dramatic change in the uniform asperity slip model, which does not reflect reality.Derived source slip models indicate that the slip on the asperity gradually decreases from the central zone to the boundary.
To address the spatial variation of the slip on the asperity, a nonuniform asperity slip model is proposed, in which the slip on the asperity gradually changes, as shown in Fig. 3.The gray area filled with lines represents subfaults on the asperity adjacent to the background area, and the white area filled with lines represents subfaults in the background area adjacent to the asperity.There are two asperities in this model: a large one and a small one.For a fault with n subfaults of the same size, when the average slip u as of the asperities and the average slip u = 10 1.5M w +9.1 µA .
(6) u b of the background on the fault have been obtained, the dislocation amount of subfaults can be determined by the following method.
For an asperity with n as subfaults, there are n aso subfaults adjacent to the background area, with an average dislocation u aso , and n asi subfaults not adjacent to the background region, with an average dislocation u asi .u aso and u asi can be determined as follows: Assuming the shear modulus of the entire fault plane is constant, Eq. ( 7) takes the following form: Moreover, the average slip u asi is assumed to be two times of u aso The dislocation of each subfault in the adjacent background area on the asperity u asoi (1 ≤ i ≤ n aso ) and the dislocation of the other subfaults on the asperity u asij (1 ≤ j ≤ n asi ) have the following relationships: Since the shear modulus of the entire fault plane is constant, and the subfaults have the same areas, Eqs.(10)  and (11) can be rewritten as follows: To account for the variation in u asoi , u asoi is assumed to take a random value in the range between 0.5u aso and 1.5u aso .u asij follows the same rule.The slip on the asperity can then be determined by this assumption and Eqs.(8), ( 9), (12), and (13).The slip on the background area can be determined using the same method.Using the above method, the nonuniform slip on the source can be obtained, which considers the general characteristics of the inversed source models.

Surface deformation of Loma Prieta earthquake by nonuniform slip model
In this section, the 1989 M w 6.95 Loma Prieta earthquake is selected as the target event.The main source parameters of the Loma Prieta earthquake are presented in Table 1.To validate the proposed nonuniform slip model, three different slip distribution models are constructed as shown in Fig. 4. All three source models have the same seismic moments.Model 1 is an inverted slip distribution based on seismic data 27 .In Model 2, the dislocations on the asperities and the background are uniform.In Model 3, the heterogeneous slip distribution is determined using the method proposed in Section "Nonuniform source slip model".The surface deformations of the 1989 Loma Prieta earthquake are calculated using these three models.(7)  n as µu as A sub =n aso µ 1 u aso A sub + n asi µ 2 u asi A sub .
(8) n as u as =n aso u aso + n asi u asi .
(9) u aso = 0.5u asi .(10)   n aso i=1 µA sub u asoi = n aso µA sub u aso , (11)   n asi j=1 µA sub u asij = n asi µA sub u asi , (12)   n aso i=1 u asoi = n aso u aso , The surface deformation calculated using Model 1 is used as the reference.The surface deformation difference between Models 2 and 1 is denoted as D m21 , and D m31 follows the same rule.Figs.S1 and S2 display the surface displacement differences D m21 and D m31 in three directions, respectively.The thick gray solid line is the fault line, and the origin of the coordinates is located at the center point of the fault line.The surface deformation difference D m31 is smaller than D m21 , indicating that the surface deformation calculated by Model 3 is closer to the result of Model 1.The areas with the surface deformation difference larger than 0.04 m and larger than 0.08 m are compared in Fig. 5.The yellow bars represent the results for D m21 , and the green bars with dashed lines represent the results for D m31 .The areas of surface deformation difference are compared in three directions, which are represented as u x , u y , and u z .As shown in Fig. 5, the values of the yellow bars are all larger than those of the green bars, indicating that Model 3 with nonuniform asperity slip achieves better performance than Model 2 with uniform asperity slip.

Surface deformation of Menyuan Ms6.9 earthquake
On January 8, 2022, a magnitude 6.9 earthquake occurred in Menyuan County, Qinghai Province.The hypocenter of the Menyuan M s 6.9 earthquake was located at 37.77°N, 101.26°E, with a depth of 10 km.This event injured several people and caused damage of infrastructure.The main source parameters of the Menyuan M s 6.9 earthquake are summarized in Table 2.
The inverted dislocation distribution on the source plane is shown in Fig. S3 (https:// earth quake.usgs.gov/).Fig. S4 is the inverted slip model of Menyuan earthquake by Zhang et al. 38 .And the heterogeneous slip model inverted by Zhang et al. 38 is also referenced to build up the proposed nonuniform slip model, as shown in Fig. 6.
Figure 7 shows the surface deformation induced by the Menyuan M s 6.9 earthquake, which is calculated based on the source slip model in Fig. 6.The thick gray line represents the fault line, and the star marks the epicenter.
In the fault-parallel direction (Fig. 7a), the displacement is positive in the southeast direction and negative in the northwest direction.The surface deformation shows a symmetrical distribution along the fault line but in opposite directions.In the near-fault region, the average surface deformation in the fault-parallel direction is larger than 1.5 m, with a maximum value of 2.24 m near the epicenter.The surface deformation decreases to 0.15 m at a fault distance of 10 km.According to the field research by Institute of Geology, China Earthquake Administrator(https:// eq-igl.ac.cn/ zhxw/ info/ 2022/ 36632.html), the maximum dislocation is over 2 m.The results of inverted surface deformation are matches the field research.Based on the inverted surface deformation of the Liuhuangou Bridge near the Daliang tunnel conducted by Liu et al. 39 , our results also show a close correlation with their findings.
In the fault-perpendicular direction (Fig. 7b), the surface displacement is negative in the northeast direction and displays a symmetric distribution.The maximum surface deformation is 0.36 m at a site with a fault distance of 1.5 km.Near the epicenter, the surface deformation is approximately 0 m.The vertical surface deformation (Fig. 7c) is remarkably smaller than that in the horizontal direction and shows a centrosymmetric distribution along the fault line.The maximum uplift deformation is 0.99 m, and the maximum surface subsidence is 0.15 m near the fault.Surface deformation in the three directions corresponds to the general characteristics of a strikeslip fault.

Seismic response of the Daliang tunnel caused by spatial varied internal displacement
The earthquake damage investigation after the 2022 M s 6.9 Menyuan earthquake shows that the Daliang tunnel of the Lanxin railway was damaged due to fault slip.In this section, the response of the Daliang Tunnel is estimated using a continuous-discrete coupling numerical simulation method.FLAC 3D and PFC 3D programs are used to perform the discrete-continuous coupling simulations.The spatially varied surface deformation calculated in Section "Surface deformation of Menyuan Ms6.9 earthquake" is used as the input seismic load.

Description of the Daliang tunnel and discrete-continuous coupling model
The dip angle of the fault was 88°(https:// earth quake.usgs.gov/) and the width of the fault fracture zone was 70 m.The discrete-continuous coupling model with surrounding rocks and a fault zone is 200 m wide, 500 m high, and 400 m in the longitudinal direction.The coupled discrete-continuous model of the Daliang tunnel and surrounding rock mass is shown in Fig. 8a.The Daliang tunnel has a radius of 7.5 m, a lining of 0.5 m, a longitudinal length of 400 m, and a depth of 400 m.The dimensions of the tunnel are detailed in Fig. 8b.The boundary conditions of the model involve applying the internal deformation calculated in Section "Surface deformation  of Menyuan Ms6.9 earthquake" to the top, bottom, left, and right boundaries of the hanging wall, as well as to part of the fracture zone.The rest of the model are fixed to ensure stability.FLAC 3D and PFC 3D programs are used to perform the discrete-continuous coupling simulations.The surrounding rock and fault fracture zone are modeled as discrete particles, and the parameters of the discrete particles are listed in Table 3.The tunnel structure is modeled as continuous elements, the behavior of which is described by the Mohr-Coulomb plastic model.The tunnel parameters are listed in Table 4.These parameters are determined based on the research of Zhang et al. 40 .The coupling process between the discrete and continuous Table 2. Source parameters of the M S 6.9 Menyuan earthquake.

Parameter Value
Length (km) 25 elements is illustrated in Fig. S5, which is described by Cai et al. 41 and Ma et al. 25 .The internal deformation calculated in Section "Surface deformation of Menyuan Ms6.9 earthquake" is employed as the input seismic load.

Response of the Daliang tunnel by discrete-continuous coupled numerical simulation
Figure 9 shows the longitudinal strain of the lining, which is considered positive during tensile deformation.The region with a large longitudinal strain of the lining is concentrated around the fault zone within a range of 40 m.The maximum tensile strain of the right side wall exceeded 5.5 × 10 3 με, and the maximum tensile strain of the left side wall is approximately 5.5 × 10 3 με.The local deformation in the lining has a range of 150 m in the fault zone.According to the elastic modulus of the material, the ultimate compressive strain and tensile strain of the material are 3 × 10 3 με and 1 × 10 2 με, respectively.The horizontal displacements of the left and right side walls are shown in Fig. 10a and b, respectively.The northward displacement of the tunnel is taken as positive, and the maximum relative dislocation of the tunnel is approximately 1.09 m.Because the fault is set as a left-lateral strike-slip fault in the numerical model, there is no obvious vertical deformation of the tunnel.The maximum uplift and maximum subsidence of the tunnel are 0.1 m and 0.2 m, respectively.
On-site seismic damage investigation shows that the tunnel is severely dislocated in the fault zone by Zhang et al. 38 and the Institute of Geology, China Earthquake Administrator(https:// eq-igl.ac.cn/ zhxw/ info/ 2022/ 36632.html).The maximum horizontal dislocation of the Daliang tunnel are over 2.5 m, respectively.The rail exhibits an S-shaped deformation, which is represented by the red curve in Fig. 15.The maximum extruded dislocation of the left side wall and right side wall are 2.16 m and 2.17 m, respectively.The observed results satisfy the characteristics of left-lateral strike-slip faults.
The response of the Daliang tunnel calculated by discrete-continuous coupled numerical simulation is generally consistent with the on-site seismic damage investigation.However, the maximum dislocation in the numerical model is smaller than the observed result, particularly in the vertical direction.This discrepancy can be attributed to two reasons.The first is the difference between the parameters in the numerical model and the real surrounding rocks in both the internal deformation simulation and the response analysis of the tunnel.The second reason is the effect of seismic waves on the tunnel, which is not considered in the numerical model.During an earthquake, both seismic waves and dislocations affect the response of the structures.In this section, we focus only on the response induced by ground deformation, which causes an underestimation of the deformation of the tunnel after an earthquake.Despite the above-mentioned discrepancy, the regional and overall stress and deformation characteristics of the tunnel model satisfy the real status of the Daliang tunnel after the M s 6.9 Menyuan earthquake.The numerical simulation results show that the discrete-continuous coupling numerical model is valid for the analysis of tunnel passing through fault zones.

Discussion-effects of different fault conditions on the response of tunnels
Different active faults have varying parameters that cause different ground deformation distributions.To investigate the effects of different fault parameters on the response of tunnels through active faults, a discrete-continuous coupling numerical simulation is performed under different fault mechanisms, fault slips, rupture fault widths, fault dip angles, and location of the asperity.3 and Table 4.The numerical model of the strike-slip fault and tunnel is shown in Fig. 11.The models of other three fault mechanisms are the same as that of strike-slip fault, the only difference is the slip direction.Figure 12 shows the different fault mechanism.
Section "Description of the Daliang tunnel and discrete-continuous coupling model" elucidates the principles of force and velocity transfer at the interface between FLAC3D and PFC3D during coupling, where contact forces are linked to displacements through stiffness in the contact surfaces.To validate the effectiveness of the coupled model, a normal fault model with a 60° dip angle was employed, examining displacements in both discrete and continuous elements.The model parameters include a 0.6 m fault slip and a 20 m fracture zone width, as depicted in Fig. 11.The resulting vertical displacement of the tunnel crossing the fault is illustrated in Fig. 13.The displacement contour map shows that the fault dip induces approximately 0.52 m of vertical displacement in the hanging wall with a 0.6 m fault slip.The tunnel moves downward along with the fault and deforms near the fracture zone.The uniform displacement scale for the fault and tunnel in the map indicates strong continuity between the fault's discrete particles and the tunnel's continuous structure.
Figure 14 displays the longitudinal strain of the crown, invert, left side wall, and right side wall under four different fault mechanisms.For the normal and reverse faults, the peak longitudinal strains of the tunnel appear   at the crown and invert.For the strike-slip faults, the peak longitudinal strains of the tunnel appear at the left side and right side walls.These results indicate that the distribution of the peak strain in the tunnel is governed by the fault mechanism.For tunnels passing through active fault zones, the fault mechanism should be considered in the seismic design of tunnels.The peak deformation in the tunnel is mainly concentrated in the vicinity of the fault fracture zone.In these cases, the tensile strain in the tensile part of the tunnel exceeds the ultimate tensile strain of the concrete, indicating that the damage to the tunnel is caused by tensile deformation.

Effect of the fault slip on the tunnel
The fault slip increases with the earthquake magnitude.To investigate the effect of fault slips on the response of the tunnel through active faults, the fault slip is set to be 0.2 m, 0.6 m, 0.8 m, and 1 m.The fault is assumed to be a reverse fault with a dip angle of 60° and a fracture zone width of 20 m.The discrete-continuous coupling model is 60 m wide, 60 m high, and 200 m in the longitudinal direction.The buried depth of the tunnel is 26 m, and its calculated length is 200 m.The parameters of the tunnel, surrounding rock, and fault fracture zone are the same as those in Section "Description of the Daliang tunnel and discrete-continuous coupling model", as listed in Tables 3 and 4.
Figures 15a and b show the longitudinal stress and strain of the crown under different fault slips.The stress and strain of the tunnel are considered to be positive during tensile deformation.The longitudinal stress and strain generally increase with the fault slip.The maximum longitudinal tensile stress at the crown appears on the hanging wall, approximately 10 m from the center of the fault fracture zone.The maximum longitudinal compressive stress occurs on the footwall, at a distance of 40 m from the center of the fault fracture zone.The compressive area of the crown is larger than the tensile area.When the slip of the fault reaches 0.4 m, the maximum tensile stress of the tunnel crown exceeds 20 MPa.In the longitudinal strain curve, the tensile strain is positive and the www.nature.com/scientificreports/compressive strain is negative.The strain curves display similar trends to the stress curves.When the fault slip reaches 0.4 m, the maximum tensile strain of the tunnel exceeds 1 × 10 3 με, which obviously reached the ultimate tensile strain of concrete, while the compression area is relatively safe.

Effect of the rupture fault width on the tunnel
According to a previous report, underground structures would experience more serious damage in regions with varied geological conditions 42 .The fault fracture zone has different geological parameters and characteristics from those of the surrounding rock, which may affect the response of the tunnel through the fault zones.In this section, the width of the fault fracture zone is set to 10 m, 20 m, 30 m, and 40 m, respectively.The fault is assumed to be a reverse fault with a dip angle of 60°.The discrete-continuous coupling model is 60 m wide, 60 m high, and 200 m in the longitudinal direction.The buried depth of the tunnel is 26 m, and its calculated length is 200 m.The dislocation of the fault is 0.6 m.The other parameters of the tunnel, surrounding rock, and fault fracture zone are the same as those in Section "Description of the Daliang tunnel and discrete-continuous coupling model", as listed in Table 3.
Figure 16 shows the longitudinal stress and strain at the crown for different widths of the fault fracture zones.The maximum longitudinal stress and strain decrease as the width of the fault fracture zone increase.The maximum longitudinal tensile stress at the crown appears on the hanging wall, approximately 20 m from the center The strength of the fault fracture zone is generally lower than that of the surrounding rock, which causes damage to the tunnel concentrated in this region.With a larger fault fracture zone width, the tunnel damage would be distributed in a larger area, but the maximum values of the stress and strain would be smaller.

Effect of the dip angle on the tunnel
In this section, the dip angle of the fault is set to 30°, 45°, and 60° to investigate the effect of the dip angle on the response of the tunnel.The fault is assumed to be a reverse fault with a fracture zone width of 20 m.The discrete-continuous coupling model has the same size as that in Section "Description of the Daliang tunnel and discrete-continuous coupling model".The buried depth of the tunnel is 26 m, and its calculated length is 200 m.The dislocation of the fault is 0.6 m.The other parameters of the tunnel, surrounding rock, and fault fracture zone are the same as those in Sect.3.1, as listed in Tables 3 and 4. www.nature.com/scientificreports/ Figure 17 shows the longitudinal stress and strain at the crown under different dip angles.As the fault dip angle increases from 30° to 60°, the maximum stress and strain of the crown gradually increase.Because the fault is a reverse fault and the fault slip is constant, as the dip angle of the fault increases, the vertical component of the fault slip increases, resulting in stronger compression of the tunnel crown and invert.The horizontal component of the dislocation has little effect on the longitudinal stress of the tunnel; therefore, under the condition of the same fault slip, a larger fault dip angle will have a more adverse effect on the tunnel.
The above numerical simulation results indicate that fault slips can significantly affect the response of the tunnel.A tunnel through an active fault with a wider fault fracture zone and smaller dip angle would experience less damage.Moreover, the distributions of the maximum stress and strain change with different fault conditions.

Effect of the location of the asperity on the tunnel
The location of the rupture initiation point is a critical parameter in simulating ground motion, as it influences both the time delay and the rupture propagation direction, thereby significantly impacting the simulated ground motion of the scenario earthquake 43,44 .Ground surface displacement simulations are based on Okada's models, which do not incorporate temporal factors.Consequently, while the rupture initiation point profoundly affects the ground motion of the scenario earthquake, its impact on surface displacement is minimal.In this section, the influence of the location of the asperity on the tunnel's response is investigated.The properties of the Menyuan earthquake are used to develop the nonuniform asperity model.We adjust the asperity's location and propose three different source models, which are displayed in Fig. S6.Subsequently, the ground deformation for each source model is calculated, shown in Fig. S7 with the x dislocation along the fault.As the large asperity shifts to the right, the peak value of the x dislocation also moves rightward.This deformation is then applied to the tunnel model to study its influence on the tunnel structure.The fault is modeled as a leftlateral strike-slip fault.
Figure 18 displays the longitudinal stress on the right and left side walls under three different asperity locations.As the asperity moves rightward along the fault, the displacement within the tunnel space decreases.According to Section "Effect of the fault mechanism on the tunnel", the focus is on the mechanical response of the side wall when the fault is left-lateral strike-slip.The results indicate that the peak longitudinal stress on both side walls decreases as the asperity location shifts to the right.Consequently, the closer the tunnel is to the large asperity, the higher the stress experienced by the tunnel.

Figure 2 .
Figure 2. Asperity model with uniform slip.The gray areas represent the large and small asperities, respectively.

n asi j=1 uFigure 3 .
Figure 3. Asperity model with nonuniform slip.The gray area filled with lines represents subfaults on the asperity adjacent to the background area, and the white area filled with lines represents subfaults in the background area adjacent to the asperity.

Figure 4 .
Figure 4. Three different slip model of the Loma Prieta earthquake (unit: cm), (a) Model 1: slip model of the Loma Prieta earthquake inverted by Somerville et al. 27 , (b) Model 2: uniform asperity model of the Loma Prieta earthquake, and (c) Model 3: nonuniform asperity model of the Loma Prieta earthquake.

u x 8cm u x 4cm u y 4cm u y 8cm u z 4cm u z 8cm Figure 5 .
Figure 5. Histogram of the influence range of displacement residual between different dislocation modes and the real situation.Areas with the surface deformation difference larger than 0.04 m and larger than 0.08 m.The yellow bars and green bars with dash line represent the results of D m21 and D m31 , respectively.u x , u y , and u z represent the surface difference in three different directions, respectively.

Figure 8 .
Figure 8. Numerical model of Daliang tunnel and surrounding rocks, (a) coupled discrete-continuous model of the Daliang tunnel and surrounding rock mass and (b) model of the Daliang tunnel.

Figure 9 .
Figure 9. Longitudinal strain of the Daliang tunnel, (a) right side wall and (b) left side wall.

Figure 10 .
Figure 10.Horizontal deformation of the Daliang tunnel, (a) right side wall and (b) left side wall.The red solid curves are the schematic diagram of the real deformation of the tunnel.

Figure 11 .
Figure 11.Discrete-continuous coupling numerical model of the strike-slip fault and tunnel.

Figure 12 .
Figure 12.Schematic diagram of the fault mechanism.The pointer represents the direction of movement of the hanging wall.

Figure 15 .
Figure 15.Response of the crown under different fault slips, (a) longitudinal stress and (b) longitudinal strain.

Figure 16 .
Figure 16.Response of the crown under different fracture zone widths, (a) longitudinal stress and (b) longitudinal strain.

Figure 17 .
Figure 17.Response of the crown under different fault dips, (a) longitudinal stress and (b) longitudinal strain.

Figure 18 .
Figure 18.Longitudinal strain curve of tunnel under different asperities, (a) right side wall longitudinal stress and (b) left side wall longitudinal stress.

Table 1 .
Source parameters of the Loma Prieta earthquake.

Table 3 .
Parameters of discrete particles.

Table 4 .
Parameters of tunnel.